fabs fraction 1

Average Error: 1.7 → 0.3

Time: 12.9s

Precision: binary64

Cost: 8648

Math

\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

↓

\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ t_1 := t_0 - \frac{x}{y} \cdot z\\ t_2 := \left|\frac{z}{\frac{y}{x}} + \frac{-4 - x}{y}\right|\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+114}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-133}:\\ \;\;\;\;\left|t_0 - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y))
        (t_1 (- t_0 (* (/ x y) z)))
        (t_2 (fabs (+ (/ z (/ y x)) (/ (- -4.0 x) y)))))
   (if (<= t_1 -4e+114)
     t_2
     (if (<= t_1 1e-133) (fabs (- t_0 (/ x (/ y z)))) t_2))))

C

double code(double x, double y, double z) {
	return fabs((((x + 4.0) / y) - ((x / y) * z)));
}

↓

double code(double x, double y, double z) {
	double t_0 = (x + 4.0) / y;
	double t_1 = t_0 - ((x / y) * z);
	double t_2 = fabs(((z / (y / x)) + ((-4.0 - x) / y)));
	double tmp;
	if (t_1 <= -4e+114) {
		tmp = t_2;
	} else if (t_1 <= 1e-133) {
		tmp = fabs((t_0 - (x / (y / z))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((((x + 4.0d0) / y) - ((x / y) * z)))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y
    t_1 = t_0 - ((x / y) * z)
    t_2 = abs(((z / (y / x)) + (((-4.0d0) - x) / y)))
    if (t_1 <= (-4d+114)) then
        tmp = t_2
    else if (t_1 <= 1d-133) then
        tmp = abs((t_0 - (x / (y / z))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return Math.abs((((x + 4.0) / y) - ((x / y) * z)));
}

↓

public static double code(double x, double y, double z) {
	double t_0 = (x + 4.0) / y;
	double t_1 = t_0 - ((x / y) * z);
	double t_2 = Math.abs(((z / (y / x)) + ((-4.0 - x) / y)));
	double tmp;
	if (t_1 <= -4e+114) {
		tmp = t_2;
	} else if (t_1 <= 1e-133) {
		tmp = Math.abs((t_0 - (x / (y / z))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	return math.fabs((((x + 4.0) / y) - ((x / y) * z)))

↓

def code(x, y, z):
	t_0 = (x + 4.0) / y
	t_1 = t_0 - ((x / y) * z)
	t_2 = math.fabs(((z / (y / x)) + ((-4.0 - x) / y)))
	tmp = 0
	if t_1 <= -4e+114:
		tmp = t_2
	elif t_1 <= 1e-133:
		tmp = math.fabs((t_0 - (x / (y / z))))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	return abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z)))
end

↓

function code(x, y, z)
	t_0 = Float64(Float64(x + 4.0) / y)
	t_1 = Float64(t_0 - Float64(Float64(x / y) * z))
	t_2 = abs(Float64(Float64(z / Float64(y / x)) + Float64(Float64(-4.0 - x) / y)))
	tmp = 0.0
	if (t_1 <= -4e+114)
		tmp = t_2;
	elseif (t_1 <= 1e-133)
		tmp = abs(Float64(t_0 - Float64(x / Float64(y / z))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = abs((((x + 4.0) / y) - ((x / y) * z)));
end

↓

function tmp_2 = code(x, y, z)
	t_0 = (x + 4.0) / y;
	t_1 = t_0 - ((x / y) * z);
	t_2 = abs(((z / (y / x)) + ((-4.0 - x) / y)));
	tmp = 0.0;
	if (t_1 <= -4e+114)
		tmp = t_2;
	elseif (t_1 <= 1e-133)
		tmp = abs((t_0 - (x / (y / z))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -4e+114], t$95$2, If[LessEqual[t$95$1, 1e-133], N[Abs[N[(t$95$0 - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)) < -4e114 or 1.0000000000000001e-133 < (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))

   1. Initial program 0.2

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   2. Applied egg-rr 0.3

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{z}{\frac{y}{x}}}\right| \]

   if -4e114 < (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)) < 1.0000000000000001e-133

   1. Initial program 3.8

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   2. Applied egg-rr 0.2

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x}{\frac{y}{z}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq -4 \cdot 10^{+114}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}} + \frac{-4 - x}{y}\right|\\ \mathbf{elif}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq 10^{-133}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}} + \frac{-4 - x}{y}\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	12.6
Cost	7248

\[\begin{array}{l} t_0 := \left|\frac{x}{y} \cdot z\right|\\ t_1 := \left|\frac{-4 - x}{y}\right|\\ \mathbf{if}\;z \leq -4.9878821537803864 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+220}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 2
Error	0.4
Cost	7240

\[\begin{array}{l} t_0 := \left|\frac{x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{if}\;x \leq -2 \cdot 10^{+60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.4
Cost	7240

\[\begin{array}{l} t_0 := \frac{z}{\frac{y}{x}}\\ \mathbf{if}\;x \leq -6.860711909990143 \cdot 10^{-70}:\\ \;\;\;\;\left|t_0 + \frac{-4 - x}{y}\right|\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} - t_0\right|\\ \end{array} \]

Alternative 4
Error	20.3
Cost	7116

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.136472908941336 \cdot 10^{-98}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	20.4
Cost	7116

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.136472908941336 \cdot 10^{-98}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	10.8
Cost	7112

\[\begin{array}{l} t_0 := \left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{if}\;x \leq -0.0165:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.136472908941336 \cdot 10^{-98}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	10.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0165:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 1.136472908941336 \cdot 10^{-98}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \]

Alternative 8
Error	10.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0165:\\ \;\;\;\;\left|\frac{x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 1.136472908941336 \cdot 10^{-98}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \]

Alternative 9
Error	18.7
Cost	6856

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	47.2
Cost	6592

\[\left|\frac{x}{y}\right| \]

Reproduce

herbie shell --seed 2022318 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))